Anomaly-induced Thermodynamics in Higher Dimensional AdS/CFT Gim - - PowerPoint PPT Presentation

Anomaly-induced Thermodynamics in Higher Dimensional AdS/CFT

Gim Seng Ng

McGill University

Northeast Gravity Workshop, 23rd April 2016

Based on : 1311.2940, 1407.6364 and 1505.02816 
 (with T. Azeyanagi, R. Loganayagam and M. J. Rodriguez)

Motivations

BHs have entropy ! Entropy-matching … Often extremal or/and susy… How about non-extremal/ non-SUSY finite temperature entropy matching? BTZ entropy is reproduced by the universal Cardy’s formula Higher-dimensional generalizations of Cardy’s formula? Higher-dimensional AdS/CFT “Cardy entropy-matching”?

Outline

Chiral Half of Cardy’s Formula in Replacement Rule from 2n 2n+12n

chiral half of Cardy

Consider a 2d CFT on a circle of radius R Let us put it at finite temperature and rotation/boost At high temperature:

R Time

T 1/R

Gravitational Anomaly (``Chiral Half’’/anomalous-part) Weyl Anomaly

SCardy ≈ 2πR 1 − R2Ω2 cR + cL 24 (4πT)

• +

2πR2Ω 1 − R2Ω2 cR − cL 24 (4πT)

chiral HALF of Cardy

SCFT2,anom ≈ 2πR2Ω 1 − R2Ω2 cR − cL 24 (4πT)

• Generalizations to higher-d CFT (on sphere)

Replacement rule To understand the replacement rule, we need to review the following two things: T=0 anomalies Anomalous hydrodynamics

[Surowka, Loganayagam, Jensen, Yarom,…]

ANOMAL Y INFLOW

Anomalies are captured by Chern-Simons terms Append an extra auxiliary direction. The (2n+1) theory is anomaly-free, but with Chern-Simons terms Non-conservation of the (2n)-theory is captured by the ``inflow’’ of charges into the extra auxiliary direction:

μJμ|QFT2n j⊥

j⊥

QFT2n

j⊥ ∼ Panom F Panom = dICS ICS

Panom F R [Callan and Harvey]

anomalous hydro

Hydro: effective long-wavelength description:

{ uα, T, μ, . . .}

• Aα, gαβ, . . .
• Hydro derivative expansion:

Leading anomalous contribution: parity-odd vorticity :

Jα = quα + . . . +(Jα)anom + . . . Tαβ = Euαuβ + p

• gαβ + uαuβ
• + . . . +(Tαβ)anom + . . .

Vα εαβ...uβ (u)n−1

...

anomalous hydro

(Jα)anom = −∂F[T, μ] ∂μ Vα + . . .

(Tαβ)anom =

• F − μ

∂F ∂μ

• T

− T ∂F ∂T

• μ
• uαVβ + Vαuβ
• + . . .

is like an anomalous Gibbs free energy

Recall that standard relations:

F

Q = −∂G ∂μ , E = G − μ ∂G ∂μ

• T

− T ∂G ∂T

• μ

, S = − ∂G ∂T

• μ

Jα = quα + . . . +(Jα)anom + . . . Tαβ = Euαuβ + p

• gαβ + uαuβ
• + . . . +(Tαβ)anom + . . .

Sanom = − F T

• μ
• CFTVμdxμ = −

F T

• μ
• i

(2πR2Ωi)

anomalous hydro

(Jα)anom = −∂F[T, μ] ∂μ Vα + . . . (Tαβ)anom =

• F − μ

∂F ∂μ

• T

− T ∂F ∂T

• μ
• uαVβ + Vαuβ
• + . . .

Question: What is ?

F

Sanom = − F T

• μ
• CFTVμdxμ = −

F T

• μ
• i

(2πR2Ωi)

chiral HALF of Cardy

Example: 2d CFT Cardy’s formula

Sanom = −∂F ∂T (2πR2Ω) SCFT2,anom ≈ 2πR2Ω cR − cL 24 (4πT)

• F = − cR − cL

2(2π)242(2πT)2 = cg [R2]

• trR2→2(2πT)2

anomaly polynomial for 2d grav. anomaly

Replacement rule OR chiral half of cardy’s formula

[Surowka, Loganayagam, Jensen, Yarom,…]

(Jα)anom = −∂F[T, μ] ∂μ Vα + . . . (Tαβ)anom =

• F − μ

∂F ∂μ

• T

− T ∂F ∂T

• μ
• uαVβ + Vαuβ
• + . . .

Vα εαβ...uβ(u)n−1

Sanom = − F T

• μ
• CFTVμdxμ = −

F T

• μ
• i

(2πR2Ωi)

F = Panom[F, R]|F→μ,[R2k]→2(2πT)2k

REPLACEMENT RULE

2d: Panom = cAF2 + cg[R2],

F = cAμ2 + cg2(2πT)2

4d: Panom = cAF3 + cMF[R2],

F = cAμ3 + cMμ × 2(2πT)2

6d: Panom cg[R4],

F = cg2(2πT)4

(Jα)anom = −∂F[T, μ] ∂μ Vα + . . . (Tαβ)anom =

• F − μ

∂F ∂μ

• T

− T ∂F ∂T

• μ
• uαVβ + Vαuβ
• + . . .

Sanom = − F T

• μ
• CFTVμdxμ = −

F T

• μ
• i

(2πR2Ωi)

Outline

Chiral Half of Cardy’s Formula in Replacement Rule from


• Testing AdS/CFT in the presence of anomalies

• What bulk geometric structure captures the


replacement rule? 2n 2n+12n

A holographer ’s recipe

• 1. Find your favorite AdS bulk theories
• 2. Find the relevant AdS BH solutions
• 3. Calculate responses, charges, thermodynamics…
• 4. Match/predict CFT results

Issue 1: Need gauge-gravitational Chern-Simons terms Issue 2: Need AdS-Kerr-Newman+CS solutions 
 (with all rotations/charges turned on) Issue 3: Holographic renormalization/charges / entropy for Chern-Simons terms

step 1: favorite Ads gravity setup

Toy model: D=2n+1 Einstein-Maxwell+negative c.c.
 +CS terms Ics Equations of motion:

Rab − 1 2 (R − 2Λ) gab = 8πGN

• (TM)ab + (TH)ab
• bFab = g2

YM (JH)a

(TM)ab

JH = − Panom F

• (TH)ab = cΣ(ab)c

Σ(ab)c = −2 Panom Rab

• CS/``Hall’’ contributions:

Maxwell contribution:

Panom = dICS

step 2: fluid/gravity BH solution (no anomalies)

Gravity dual of charged rotating fluid
 (for now, without anomalies)

ds2 = −2uμdxμdr + r2 −f(r, m, q)uμuν + Pμν

• dxμdxν +

A = Φ(r, q)uμdxμ +

Pμν = gμν + uμuν

Φ(r, q) = q r2n−2

f(r, m, q) = 1 − m r2n + 1 2κq q2 r2(2n−1)

Non-trivial bulk radial dependence: Horizon values:

Φ(rH) = μ, ΦT(rH) = 2πT

ΦT(r) ≡ r2 2 df dr

step 2: fluid/gravity BH solution (with anomalies)

Gravity dual of anomalous charged rotating fluid

ds2 = −2uμdxμdr + r2 −f(r, m, q)uμuν + Pμν

• dxμdxν +

A = Φ(r, q)uμdxμ +

Leading contributions from the CS-terms :


+gV(r, m, q)(uμVν + uνVμ)dxμdxν + . . . +aV(r, m, q)Vμdxμ + . . .

gV, aV

Rab − 1 2 (R − 2Λ) gab = 8πGN

• (TM)ab + (TH)ab
• bFab = g2

YM (JH)a

Bulk replacement rule:

TH ∼ ∂2

r

∂G ∂ΦT

• JH ∼ ∂r

∂G ∂Φ

• G ≡ Panom
• F → Φ, [R2k] → 2Φ2k

T

• ΦT(r) ≡ r2

2 df dr

step 3&4: currents and replacement rule

• Tαβ
• anom =
• T

αβ

• anom =
• G − Φ ∂G

∂Φ − ΦT ∂G ∂ΦT

• r=rH
• uαVβ + uβVα
• G(r = rH) = F[F → μ, [R2k] → 2(2πT)2k]

Bulk replacement rule metric corrections boundary stress-tensor/currents

(Jα)anom ∼ gμα (Frμ)anom | = − ∂G ∂Φ

• r=rH

Vα

Bulk replacement rule-> boundary replacement rule !!

step 3&4: entropy

Tachikawa Formula (see also Solodukhin and Bonora-Cvitan-Prester-Pallua-Smolic)
 (note: NOT Wald’s formula applied to Ics)

SCS =

• ∞
• k=1

(8πk)ΓNR2k−2

N

∂Panom ∂[R2k]

Panom = dIcs, ΓN, RN =

Applying this to the fluid/gravity metric we found gives

Sanom = − F T

• μ
• CFTVμdxμ = −

F T

• μ
• i

(2πR2Ωi)

Computations are opaque, long and tedious … 
 (what bulk structures imply the replacement for entropy?)

summary, current and future work

Higher-dimensional chiral-half of Cardy’s formula 
 (Replacement rule) Construct (anomalous) fluid/gravity solutions in the bulk for the Einstein-Maxwell-CS theory Found ``bulk replacement rule’’ which implies the boundary replacement rule Current/Future:

Time-dependence (non-stationary) and/or higher-order? More realistic AdS/CFT setup … add matter and etc 
 (should not alter the conclusions) Anomaly-induced entanglement entropy [Azeyanagi-Loganayagam-Ng, 1507.02298]